For \(k\in \lbrace 2, \ldots , 2n+2\rbrace \) 
let \(\Delta _k \in B_k\)  be the longest reduced positive braid in the group \(B_k\)  of braids with \(k\)  strands.
It is defined inductively by \(\Delta _2 = \sigma _1\)  and
\(\Delta _{k+1} = (\sigma _1 \sigma _2 \cdots \sigma _k) \, \Delta _k\)  for \(k\ge 2\) .
It is well known (see [1]} or [2]}) that its square \(\Delta _k^2\)  generates the center of \(B_k\) .
